Diagonal Matrix : Definition, Inverse, Properties, Examples, Anti

Generally, a matrix in which all the elements, except “Leading Diagonal” are ZERO is called the Diagonal Matrix(DM). And of course, it is mandatory that, the matrix should be a Square Matrix of order n×n. If the matrix is not square. Then the Leading Diagonal is not the diagonal of the matrix. In short, this affects the properties of the DM. There exist some anti diagonal matrices, the inverse of the matrices, examples which are explained further.

What is a Diagonal Matrix?

A diagonal matrix is defined as the square matrix in which all the elements are zero except the leading diagonal elements. Sometimes people may say this leading diagonal is the principal diagonal.

By definition you can say, ( aij = 0 when i ≠ j ) means aij = 0, wherever ‘ i ‘ is not equal to ‘ j ‘. Generally, in any Matrix, if this condition is met, it is called DM. Obviously, a11, a22, and a33 should be non-ZERO for a 3×3 matrix.

{ \begin{bmatrix}a_{11} &0 &0 \\0 & a_{22} &0 \\0 & 0 &a_{33} \end{bmatrix} }_{3 × 3}

Here, “ i ” corresponds to a row. whereas “ j ” corresponds to a column.

Further, there are some examples,

  • Scalar matrix –

Scalar Matrix=constant × Identity Matrix.

Scalar Matrix=K× {\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}  }_{2 × 2}
Scalar Matrix={\begin{bmatrix}K & 0 \\0 & K \end{bmatrix}  }_{2 × 2}
  • Identity Matrix
Identity Matrix={\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}  }_{2 × 2}
  • Null Matrix –
Null Matrix={\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}  }_{2 × 2}

Before proceeding further, you need some terminologies known.

Triangular matrices

Every Square Matrix is surely having a Leading diagonal. It starts from the first element and ends on the last element. Collectively, aij is the element of the Leading diagonal but i = j.

Every square matrix form two triangular regions. One is above the leading diagonal whereas the other is below the leading diagonal.

What is an Upper triangular matrix?

{ \begin{bmatrix}1 & 2 &3 \\0 & 2 &4 \\0 & 0 &3 \end{bmatrix}  }_{3 × 3}

A matrix in which all the elements except upper triangular region elements are non-zero and lower triangular region elements is zero then it is called an upper triangular matrix.

What is a Lower triangular matrix?

 {\begin{bmatrix}1 & 0 & 0 \\1 & 2 &0 \\4 & 5 &3 \end{bmatrix}  }_{3 × 3}

A matrix in which all the elements of the upper triangular region are zero and the lower triangular region is non-zero then it is called a Lower triangular matrix.

Note, A DM is both “Upper triangular matrix” as well as “Lower triangular matrix”.

 {\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}  }_{2 × 2}

Properties of the Diagonal Matrix (DM)

  • Additive Property – If two DM of the same order are added, the resultant will be also a Diagonal matrix. For example –
if A=  {\begin{bmatrix}3 & 0 \\0 & 2 \end{bmatrix}  }_{2 × 2} ,  B=  {\begin{bmatrix}2 & 0 \\0 & 1 \end{bmatrix}  }_{2 × 2}
A+B=  {\begin{bmatrix}3+2 & 0+0 \\0+0 & 2+1 \end{bmatrix}  }_{2 × 2}
A+B=  {\begin{bmatrix}3 & 0 \\0 & 2 \end{bmatrix}  }_{2 × 2} + {\begin{bmatrix}2 & 0 \\0 & 1 \end{bmatrix}  }_{2 × 2}
Result =  {\begin{bmatrix}5 & 0 \\0 & 3 \end{bmatrix}  }_{2 × 2}
  • Transpose property – Let matrix be A then its Transpose will be AT. By this property, A = AT. Conclusively, both the original matrix and its Transpose are the same.
ifA =  {\begin{bmatrix}5 & 0 \\0 & 3 \end{bmatrix}  }_{2 × 2} then, A^T =  {\begin{bmatrix}5 & 0 \\0 & 3 \end{bmatrix}  }_{2 × 2}
  • Commutative Property – Diagonal matrices show the commutative properties for both Multiplication and Addition. This means that, AB = BA and A+B = B+A. So, either Pre-multiply/add or Post-multiply/add Result doesn’t change. Also, the result is a DM.
if A=  {\begin{bmatrix}1 & 0 \\0 & 3 \end{bmatrix}  }_{2 × 2} ,  B=  {\begin{bmatrix}2 & 0 \\0 & 1 \end{bmatrix}  }_{2 × 2}
A×B=  {\begin{bmatrix}2+0 & 0+0 \\0+0 & 0+3 \end{bmatrix}  }_{2 × 2}
Result =  {\begin{bmatrix}2 & 0 \\0 & 3 \end{bmatrix}  }_{2 × 2}
B×A=  {\begin{bmatrix}2+0 & 0+0 \\0+0 & 0+3 \end{bmatrix}  }_{2 × 2}
Result =  {\begin{bmatrix}2 & 0 \\0 & 3 \end{bmatrix}  }_{2 × 2}
AB=BA
  • Hence every DM is a Square Matrix of order n×n.

Determinant of Diagonal Matrix

Still, the Determinant of the DM can be calculated using the normal conventional method. But at the end conclusion, you will see some tricks. Suppose we take a 3 × 3 matrix A.

A =  { \begin{bmatrix}a_{11} &0 &0 \\0 & a_{22} &0 \\0 & 0 &a_{33} \end{bmatrix} }_{3 × 3}  
|A|=a_{11}× {\begin{bmatrix}a_{22} & 0 \\0 & a_{23} \end{bmatrix} }_{2 × 2} + a_{11}× {\begin{bmatrix}0& 0 \\0 & a_{33} \end{bmatrix} }_{2 × 2} + 0× {\begin{bmatrix}0& a_{22} \\0 & 0 \end{bmatrix} }_{2 × 2}
|A| = a_{11}×a_{22}×a_{33}

Hence, we can conclude that, for a DM of any order (n×n), its Determinant will be the product of its Leading diagonal elements. Thus, if a matrix has the non-zero elements of the leading diagonal, its determinant will be non-zero.

The inverse of the Diagonal Matrix

The inverse of the diagonal matrix is the inverse of the corresponding original diagonal matrix elements. It will also be a diagonal matrix.

A =  { \begin{bmatrix}a_{11} &0 &0 \\0 & a_{22} &0 \\0 & 0 &a_{33} \end{bmatrix} }_{3 × 3}  
adjA =  { \begin{bmatrix}a_{22}a_{33} &0 &0 \\0 & a_{11}a_{33} &0 \\0 & 0 &a_{11} a_{22}\end{bmatrix} }_{3 × 3}  
|A| = a_{11}×a_{22}×a_{33}
{A} ^{-1}= \frac{1}{|A|}  \times adjA
{A} ^{-1}= \frac{1}{a_{11}a_{22}a_{33}}  \times { \begin{bmatrix}a_{22}a_{33} &0 &0 \\0 & a_{11}a_{33} &0 \\0 & 0 &a_{11} a_{22}\end{bmatrix} }_{3 × 3} 
{A} ^{-1}= { \begin{bmatrix}\frac{1}{a_{11}} &0 &0 \\0 & \frac{1}{a_{22}}  &0 \\0 & 0 &\frac{1}{a_{33}} \end{bmatrix} }_{3 × 3} 

what is an Anti-Diagonal matrix?

Lagging-anti-diagonal-in-a-matrix

Generally, A square matrix contains two diagonals, one is the Leading diagonal, and the other is the Lagging diagonal. Now, if a square matrix whose elements other than lagging diagonal are zero. this is called an Anti-diagonal Matrix.

anti-diagonal-matrix

Hence, It is like that only –

A =  { \begin{bmatrix}0 &0 &a_{13}\\0 & a_{22} &0 \\a_{31} & 0 &0\end{bmatrix} }_{3 × 3}  

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